# summation indices for multi-binomial expansion

I am trying to figure out the form of the summation indices of a multi-binomial expansion. Suppose the condensed form is

\begin{align} S &= \prod_{a=1}^{N-1} \prod_{b=a+1}^{N} (1+x_{ab}) \\ &= \prod_{a<b} (1+x_{ab}) \\ &= (1+x_{12})(1+x_{13})(1+x_{14})\cdots(1+x_{23})(1+x_{24})\cdots(1+x_{N-1,N}) \end{align}.

This can be expanded as

\begin{align} S &= (1+x_{12}+x_{13}+x_{12}x_{13})(1+x_{14})\cdots(1+x_{N-1,N})\\ &=(1+x_{12}+x_{13}+x_{14}+x_{12}x_{13}+x_{12}x_{14}+x_{13}x_{14}+x_{12}x_{13}x_{14})\cdots(1+x_{N-1,N})\\ &=1+\sum_{a<b} x_{ab} + \sum_{?} x_{ab}x_{cd} + \sum_{?}x_{ab}x_{cd}x_{ef} + \cdots + x_{12}x_{13}x_{14}\cdots{}x_{N-1,N} \end{align}.

I am trying to figure out how to express the indices of the sums of higher order (e.g., those containing question marks in the above expression). For instance, the indices in the term $\displaystyle\sum_{?} x_{ab}x_{cd}$ cannot be $\displaystyle\sum_{\substack{a<b\\c<d}} x_{ab}x_{cd}$ because this double counts terms. Nor can it be $\displaystyle\sum_{a<b<c<d} x_{ab}x_{cd}$ because this misses terms. How can I express the indices of this sum and the higher order sums in the above expression?

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